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| {{About|deformation in mechanics|the term's use in engineering|Deformation (engineering)}}
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| [[File:DeformationOfRod_plain.svg|400px|right|thumb|The deformation of a thin straight rod into a closed loop. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. In this particular case, most of the deformation in the rod is due to rigid body motions of material points in the rod.]]
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| '''Deformation''' in [[continuum mechanics]] is the transformation of a body from a ''reference'' configuration to a ''current'' configuration.<ref name=Truesdell>Truesdell, C. and Noll, W., (2004), ''The non-linear field theories of mechanics: Third edition'', Springer, p. 48.</ref> A configuration is a set containing the positions of all particles of the body.
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| A deformation may be caused by [[structural load|external loads]],<ref name=wu>H.-C. Wu, ''Continuum Mechanics and Plasticity'', CRC Press (2005), ISBN 1-58488-363-4</ref> [[body force]]s (such as [[gravity]] or [[electromagnetic force]]s), or temperature changes within the body.
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| Strain is a description of deformation in terms of ''relative'' displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the [[metric tensor]] or its dual is considered.
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| | |
| In a continuous body, a deformation field results from a [[Stress (physics)|stress]] field induced by applied [[force]]s or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by [[constitutive equations]], e.g., [[Hooke's law]] for [[Linear elasticity|linear elastic]] materials. Deformations which are recovered after the stress field has been removed are called '''elastic deformations'''. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is '''plastic deformation''', which occurs in material bodies after stresses have attained a certain threshold value known as the ''elastic limit'' or [[Yield (engineering)|yield stress]], and are the result of [[Slip (materials science)|slip]], or [[dislocation]] mechanisms at the atomic level. Another type of irreversible deformation is '''viscous deformation''', which is the irreversible part of [[viscoelasticity|viscoelastic]] deformation.
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| | |
| In the case of elastic deformations, the response function linking strain to the deforming stress is the [[Hooke's law#tensor expression|compliance tensor]] of the material.
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| {{Continuum mechanics}}
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| | |
| ==Strain==
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| {{split section|Strain measures|date=September 2013}}
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| {{see also|Stress measures}}
| |
| A strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length.
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| | |
| A general deformation of a body can be expressed in the form <math>\mathbf{x} = \boldsymbol{F}(\mathbf{X})</math> where <math>\mathbf{X}</math> is the reference position of material points in the body. Such a measure does not distinguish between rigid body motions (translations and rotations) and changes in shape (and size) of the body. A deformation has units of length.
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| | |
| We could, for example, define strain to be
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| :<math>
| |
| \boldsymbol{\varepsilon} \doteq \cfrac{\partial}{\partial\mathbf{X}}\left(\mathbf{x}-\mathbf{X}\right)
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| = \boldsymbol{F}- \boldsymbol{1}
| |
| </math>.
| |
| Hence strains are dimensionless and are usually expressed as a [[decimal|decimal fraction]], a [[percentage]] or in [[parts-per notation]]. Strains measure how much a given deformation differs locally from a rigid-body deformation.<ref>{{cite book
| |
| | last = Lubliner
| |
| | first = Jacob
| |
| | title = Plasticity Theory (Revised Edition - Berkeley Authentication Required)
| |
| | publisher = Dover Publications
| |
| | year = 2008
| |
| | url = http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf
| |
| | isbn = 0-486-46290-0}}</ref>
| |
| | |
| A strain is in general a [[tensor]] quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the ''normal strain'', and the amount of distortion associated with the sliding of plane layers over each other is the ''shear strain'', within a deforming body.<ref name=rees>{{Cite book
| |
| | last = Rees|first = David
| |
| | title = Basic Engineering Plasticity - An Introduction with Engineering and Manufacturing Applications
| |
| | publisher = Butterworth-Heinemann
| |
| | year = 2006
| |
| | url = http://books.google.ca/books?id=4KWbmn_1hcYC
| |
| | isbn = 0-7506-8025-3}}</ref> This could be applied by elongation, shortening, or volume changes, or angular distortion.<ref>"Earth."Encyclopædia Britannica from [[Encyclopædia Britannica 2006 Ultimate Reference Suite DVD]] .[2009].</ref>
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| | |
| The state of strain at a [[Continuum mechanics|material point]] of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the ''normal strain'', which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the ''shear strain'', radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.
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| | |
| If there is an increase in length of the material line, the normal strain is called ''tensile strain'', otherwise, if there is reduction or compression in the length of the material line, it is called ''compressive strain''.
| |
| | |
| ===Strain measures===
| |
| Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:
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| * [[Finite strain theory]], also called ''large strain theory'', ''large deformation theory'', deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the [[Continuum mechanics|continuum]] are significantly different and a clear distinction has to be made between them. This is commonly the case with [[elastomer]]s, [[Plasticity (physics)|plastically-deforming]] materials and other [[fluid]]s and biological [[soft tissue]].
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| * [[Infinitesimal strain theory]], also called ''small strain theory'', ''small deformation theory'', ''small displacement theory'', or ''small displacement-gradient theory'' where strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical. The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting [[Deformation (engineering)#Elastic_deformation|elastic]] behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.
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| * ''Large-displacement'' or ''large-rotation theory'', which assumes small strains but large rotations and displacements.
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| In each of these theories the strain is then defined differently. The ''engineering strain'' is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g. [[elastomers]] and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%,<ref>{{Cite book
| |
| | last = Rees|first = David
| |
| | title = Basic Engineering Plasticity - An Introduction with Engineering and Manufacturing Applications
| |
| | publisher = Butterworth-Heinemann
| |
| | year = 2006|page=41
| |
| | url = http://books.google.ca/books?id=4KWbmn_1hcYC
| |
| | isbn = 0-7506-8025-3}}</ref> thus other more complex definitions of strain are required, such as ''stretch'', ''logarithmic strain'', ''Green strain'', and ''Almansi strain''.
| |
| | |
| ====Engineering strain====
| |
| The '''Cauchy strain''' or '''engineering strain''' is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The ''engineering normal strain'' or ''engineering extensional strain'' or ''nominal strain'' ''e'' of a material line element or fiber axially loaded is expressed as the change in length Δ''L'' per unit of the original length ''L'' of the line element or fibers. The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have
| |
| | |
| :<math>\ e=\frac{\Delta L}{L}=\frac{\ell -L}{L}</math>
| |
| | |
| where <math>\ e </math> is the ''engineering normal strain,'' <math> L </math> is the original length of the fiber and <math>\ \ell </math> is the final length of the fiber. Measures of strain are often expressed in parts per million or microstrains.
| |
| | |
| The ''true shear strain'' is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The ''engineering shear strain'' is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application which sometimes makes it easier to calculate.
| |
| | |
| ====Stretch ratio====
| |
| The '''stretch ratio''' or '''extension ratio''' is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration. It is defined as the ratio between the final length ℓ and the initial length ''L'' of the material line.
| |
| | |
| :<math>\ \lambda=\frac{\ell}{L}</math>
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| The extension ratio is approximately related to the engineering strain by
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| :<math>\ e=\frac{\ell-L}{L}=\lambda-1</math>
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| This equation implies that the normal strain is zero, so that there is no deformation when the stretch is equal to unity.
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| | |
| The stretch ratio is used in the analysis of materials that exhibit large deformations, such as elastomers, which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios.
| |
| | |
| ====True strain====
| |
| The '''logarithmic strain''' ε, also called, ''true strain'' or ''Hencky strain''. Considering an incremental strain (Ludwik)
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| : <math>\ \delta \varepsilon=\frac{\delta \ell}{\ell}</math>
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| | |
| the logarithmic strain is obtained by integrating this incremental strain:
| |
| | |
| :<math>\ \begin{align}
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| \int\delta \varepsilon &=\int_{L}^{\ell}\frac{\delta \ell}{\ell}\\
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| \varepsilon&=\ln\left(\frac{\ell}{L}\right)=\ln \lambda \\
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| &=\ln(1+e) \\
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| &=e-e^2/2+e^3/3- \cdots \\
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| \end{align}
| |
| </math>
| |
| | |
| where ''e'' is the engineering strain. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path.<ref name=rees/>
| |
| | |
| ====Green strain====
| |
| {{main|Finite strain theory#Finite strain tensors|l1=Finite strain theory}}
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| The Green strain is defined as:
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| :<math>\ \varepsilon_G=\frac{1}{2}\left(\frac{\ell^2-L^2}{L^2}\right)=\frac{1}{2}(\lambda^2-1)</math>
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| | |
| ====Almansi strain====
| |
| {{main|Finite strain theory#Finite strain tensors|l1=Finite strain theory}}
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| The Euler-Almansi strain is defined as
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| | |
| :<math>\ \varepsilon_E=\frac{1}{2}\left(\frac{\ell^2-L^2}{\ell^2}\right)=\frac{1}{2}\left(1-\frac{1}{\lambda^2}\right)</math>
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| | |
| ===Normal strain===
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| [[File:2D geometric strain.svg|350px|right|thumb|Two-dimensional geometric deformation of an infinitesimal material element.]]
| |
| As with [[stress (mechanics)|stress]]es, strains may also be classified as 'normal strain' and '[[shear strain]]' (i.e. acting perpendicular to or along the face of an element respectively). For an [[isotropic]] material that obeys [[Hooke's law]], a [[normal stress]] will cause a normal strain. '''Normal strains''' produce ''dilations''.
| |
| | |
| Consider a two-dimensional infinitesimal rectangular material element with dimensions <math>dx \times dy </math>, which after deformation, takes the form of a rhombus. From the geometry of the adjacent figure we have
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| :<math>
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| \mathrm{length}(AB) = dx \,
| |
| </math> | |
| and
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| :<math>\begin{align}
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| \mathrm{length}(ab) &= \sqrt{\left(dx+\frac{\partial u_x}{\partial x}dx \right)^2 + \left( \frac{\partial u_y}{\partial x}dx \right)^2} \\
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| &= dx~\sqrt{1+2\frac{\partial u_x}{\partial x}+\left(\frac{\partial u_x}{\partial x}\right)^2 + \left(\frac{\partial u_y}{\partial x}\right)^2} \\
| |
| \end{align}\,\!</math>
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| For very small displacement gradients the squares of the derivatives are negligible and we have | |
| :<math>
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| \mathrm{length}(ab)\approx dx +\frac{\partial u_x}{\partial x}dx
| |
| </math>
| |
| The normal strain in the <math>x\,\!</math>-direction of the rectangular element is defined by
| |
| :<math>
| |
| \varepsilon_x = \frac{\text{extension}}{\text{original length}} = \frac{\mathrm{length}(ab)-\mathrm{length}(AB)}{\mathrm{length}(AB)}
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| = \frac{\partial u_x}{\partial x}
| |
| </math>
| |
| Similarly, the normal strain in the <math>y\,\!</math>-direction, and <math>z\,\!</math>-direction, becomes
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| :<math>\varepsilon_y = \frac{\partial u_y}{\partial y} \quad , \qquad \varepsilon_z = \frac{\partial u_z}{\partial z}\,\!</math>
| |
| | |
| ===Shear strain===
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| {{Infobox Physical quantity
| |
| | bgcolour =
| |
| | name = Shear strain
| |
| | image =
| |
| | caption =
| |
| | unit = [[dimensionless quantity|1]], or [[radian]]
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| | symbols = [[Gamma|γ]] or [[Epsilon|ϵ]]
| |
| | derivations = γ = [[Shear stress|τ]] / [[Shear modulus|G]]
| |
| }}
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| The engineering shear strain is defined as (<math>\gamma_{xy}</math>) the change in angle between lines <math>\overline {AC}\,\!</math> and <math>\overline {AB}\,\!</math>. Therefore,
| |
| :<math> | |
| \gamma_{xy}= \alpha + \beta\,\!
| |
| </math>
| |
| | |
| From the geometry of the figure, we have
| |
| :<math>
| |
| \begin{align}
| |
| \tan \alpha & =\frac{\tfrac{\partial u_y}{\partial x}dx}{dx+\tfrac{\partial u_x}{\partial x}dx}=\frac{\tfrac{\partial u_y}{\partial x}}{1+\tfrac{\partial u_x}{\partial x}} \\
| |
| \tan \beta & =\frac{\tfrac{\partial u_x}{\partial y}dy}{dy+\tfrac{\partial u_y}{\partial y}dy}=\frac{\tfrac{\partial u_x}{\partial y}}{1+\tfrac{\partial u_y}{\partial y}}
| |
| \end{align}
| |
| </math>
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| For small displacement gradients we have
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| :<math>
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| \cfrac{\partial u_x}{\partial x} \ll 1 ~;~~ \cfrac{\partial u_y}{\partial y} \ll 1
| |
| </math>
| |
| For small rotations, i.e. <math>\alpha\,\!</math> and <math>\beta\,\!</math> are <math>\ll 1\,\!</math> we have
| |
| <math>\tan \alpha \approx \alpha,~\tan \beta \approx \beta\,\!</math>.
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| Therefore,
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| :<math>
| |
| \alpha \approx \cfrac{\partial u_y}{\partial x} ~;~~ \beta \approx \cfrac{\partial u_x}{\partial y}
| |
| </math>
| |
| thus
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| | |
| :<math>\gamma_{xy}= \alpha + \beta = \frac{\partial u_y}{\partial x} + \frac{\partial u_x}{\partial y}\,\!</math>
| |
| By interchanging <math>x\,\!</math> and <math>y\,\!</math> and <math>u_x\,\!</math> and <math>u_y\,\!</math>, it can be shown that <math>\gamma_{xy} = \gamma_{yx}\,\!</math>
| |
| | |
| Similarly, for the <math>y\,\!</math>-<math>z\,\!</math> and <math>x\,\!</math>-<math>z\,\!</math> planes, we have
| |
| | |
| :<math>\gamma_{yz}=\gamma_{zy} = \frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y} \quad , \qquad \gamma_{zx}=\gamma_{xz}= \frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\,\!</math>
| |
| | |
| The tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, <math>\gamma\,\!</math>, as
| |
| | |
| :<math>\underline{\underline{\boldsymbol{\varepsilon}}} = \left[\begin{matrix}
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| \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\
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| \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
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| \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\
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| \end{matrix}\right] = \left[\begin{matrix}
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| \varepsilon_{xx} & \gamma_{xy}/2 & \gamma_{xz}/2 \\
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| \gamma_{yx}/2 & \varepsilon_{yy} & \gamma_{yz}/2 \\
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| \gamma_{zx}/2 & \gamma_{zy}/2 & \varepsilon_{zz} \\
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| \end{matrix}\right]\,\!</math>
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| | |
| ===Metric tensor===
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| {{main|Finite strain theory#Deformation tensors in curvilinear coordinates}}
| |
| A strain field associated with a displacement is defined, at any point, by the change in length of the [[tangent vector]]s representing the speeds of arbitrarily [[parametrized curves]] passing through that point. A basic geometric result, due to [[Maurice Fréchet|Fréchet]], [[John von Neumann|von Neumann]] and [[Pascual Jordan|Jordan]], states that, if the lengths of the tangent vectors fulfil the axioms of a [[norm (mathematics)|norm]] and the [[parallelogram law]], then the length of a vector is the square root of the value of the [[quadratic form]] associated, by the [[polarization formula]], with a [[positive definite]] [[bilinear map]] called the [[metric tensor]].
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| | |
| ==Description of deformation==
| |
| Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement. If none of the curves changes length, it is said that a [[rigid body]] displacement occurred.
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| | |
| It is convenient to identify a reference configuration or initial geometric state of the continuum body which all subsequent configurations are referenced from. The reference configuration need not be one the body actually will ever occupy. Often, the configuration at {{nowrap|1=''t'' = 0}} is considered the reference configuration, κ<sub>0</sub>('''B'''). The configuration at the current time t is the ''current configuration''.
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| | |
| For deformation analysis, the reference configuration is identified as ''undeformed configuration'', and the current configuration as ''deformed configuration''. Additionally, time is not considered when analyzing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no interest.
| |
| | |
| The components ''X''<sub>''i''</sub> of the position vector '''X''' of a particle in the reference configuration, taken with respect to the reference coordinate system, are called the ''material or reference coordinates''. On the other hand, the components ''x''<sub>''i''</sub> of the position vector '''x''' of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the ''spatial coordinates''
| |
| | |
| There are two methods for analysing the deformation of a continuum. One description is made in terms of the material or referential coordinates, called [[Continuum mechanics|material description or Lagrangian description]]. A second description is of deformation is made in terms of the spatial coordinates it is called the [[Continuum mechanics|spatial description or Eulerian description]].
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| | |
| There is continuity during deformation of a continuum body in the sense that:
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| *The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
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| *The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.
| |
| | |
| ===Affine deformation===
| |
| A deformation is called an affine deformation if it can be described by an [[affine transformation]]. Such a transformation is composed of a [[linear transformation]] (such as rotation, shear, extension and compression) and a rigid body translation. Affine deformations are also called homogeneous deformations.<ref name=Ogden>Ogden, R. W., 1984, '''Non-linear Elastic Deformations''', Dover.</ref>
| |
| | |
| Therefore an affine deformation has the form
| |
| :<math>
| |
| \mathbf{x}(\mathbf{X},t) = \boldsymbol{F}(t)\cdot\mathbf{X} + \mathbf{c}(t)
| |
| </math>
| |
| where <math>\mathbf{x}</math> is the position of a point in the deformed configuration, <math>\mathbf{X}</math> is the position in a reference configuration, <math>t</math> is a time-like parameter, <math>\boldsymbol{F}</math> is the linear transformer and <math>\mathbf{c}</math> is the translation. In matrix form, where the components are with respect to an orthonormal basis,
| |
| :<math>
| |
| \begin{bmatrix} x_1(X_1,X_2,X_3,t) \\ x_2(X_1,X_2,X_3,t) \\ x_3(X_1,X_2,X_3,t) \end{bmatrix}
| |
| = \begin{bmatrix}
| |
| F_{11}(t) & F_{12}(t) & F_{13}(t) \\ F_{21}(t) & F_{22}(t) & F_{23}(t) \\ F_{31}(t) & F_{32}(t) & F_{33}(t)
| |
| \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} +
| |
| \begin{bmatrix} c_1(t) \\ c_2(t) \\ c_3(t) \end{bmatrix}
| |
| </math>
| |
| | |
| The above deformation becomes ''non-affine'' or ''inhomogeneous'' if <math>\boldsymbol{F} = \boldsymbol{F}(\mathbf{X},t)</math> or <math>\mathbf{c} = \mathbf{c}(\mathbf{X},t)</math>.
| |
| | |
| ===Rigid body motion===
| |
| A rigid body motion is a special affine deformation that does not involve any shear, extension or compression. The transformation matrix <math>\boldsymbol{F}</math> is [[orthogonal matrix|proper orthogonal]] in order to allow rotations but no [[reflection (mathematics)|reflection]]s.
| |
| | |
| A rigid body motion can be described by
| |
| :<math>
| |
| \mathbf{x}(\mathbf{X},t) = \boldsymbol{Q}(t)\cdot\mathbf{X} + \mathbf{c}(t)
| |
| </math>
| |
| where
| |
| :<math>
| |
| \boldsymbol{Q}\cdot\boldsymbol{Q}^T = \boldsymbol{Q}^T \cdot \boldsymbol{Q} = \boldsymbol{\mathit{1}}
| |
| </math>
| |
| In matrix form,
| |
| :<math>
| |
| \begin{bmatrix} x_1(X_1,X_2,X_3,t) \\ x_2(X_1,X_2,X_3,t) \\ x_3(X_1,X_2,X_3,t) \end{bmatrix}
| |
| = \begin{bmatrix}
| |
| Q_{11}(t) & Q_{12}(t) & Q_{13}(t) \\ Q_{21}(t) & Q_{22}(t) & Q_{23}(t) \\ Q_{31}(t) & Q_{32}(t) & Q_{33}(t)
| |
| \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} +
| |
| \begin{bmatrix} c_1(t) \\ c_2(t) \\ c_3(t) \end{bmatrix}
| |
| </math>
| |
| | |
| ==Displacement==
| |
| [[File:Displacement of a continuum.svg|400px|right|thumb|Figure 1. Motion of a continuum body.]]
| |
| A change in the configuration of a continuum body results in a [[displacement field (mechanics)|displacement]]. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consist of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration <math>\ \kappa_0(\mathcal B)</math> to a current or deformed configuration <math>\ \kappa_t(\mathcal B)</math> (Figure 1).
| |
| | |
| If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the other hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero, then there is no deformation and a rigid-body displacement is said to have occurred.
| |
| | |
| The vector joining the positions of a particle ''P'' in the undeformed configuration and deformed configuration is called the [[displacement (vector)|displacement vector]] {{nowrap|1='''u'''('''X''',''t'') = ''u''<sub>''i''</sub>'''e'''<sub>''i''</sub>}} in the Lagrangian description, or {{nowrap|1='''U'''('''x''',''t'') = ''U''<sub>''J''</sub>'''E'''<sub>''J''</sub>}} in the Eulerian description.
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| A ''displacement field'' is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field. In general, the displacement field is expressed in terms of the material coordinates as
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| :<math>\ \mathbf u(\mathbf X,t) = \mathbf b(\mathbf X,t)+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J</math>
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| or in terms of the spatial coordinates as
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| :<math>\ \mathbf U(\mathbf x,t) = \mathbf b(\mathbf x,t)+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,</math>
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| where α<sub>''Ji''</sub> are the direction cosines between the material and spatial coordinate systems with unit vectors '''E'''<sub>''J''</sub> and '''e'''<sub>''i''</sub>, respectively. Thus
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| :<math>\ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}</math>
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| and the relationship between ''u''<sub>''i''</sub> and ''U''<sub>''J''</sub> is then given by
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| :<math>\ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i</math>
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| Knowing that
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| :<math>\ \mathbf e_i = \alpha_{iJ}\mathbf E_J</math>
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| then
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| :<math>\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)</math>
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| It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in {{nowrap|1='''b''' = 0}}, and the direction cosines become [[Kronecker delta]]s: | |
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| :<math>\ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}.</math>
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| Thus, we have
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| :<math>\ \mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J = x_i - X_i </math>
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| or in terms of the spatial coordinates as
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| :<math>\ \mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J =x_J - X_J</math>
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| ===Displacement gradient tensor===
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| The partial differentiation of the displacement vector with respect to the material coordinates yields the ''material displacement gradient tensor'' <math>\ \nabla_{\mathbf X}\mathbf u</math>. Thus we have:
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| :{|
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| |-
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| | <math>\begin{align}
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| \mathbf{u}(\mathbf{X},t) & = \mathbf{x}(\mathbf{X},t) - \mathbf{X} \\
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| \nabla_\mathbf{X}\mathbf{u} & = \nabla_\mathbf{X}\mathbf{x} - \mathbf{I} \\
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| \nabla_\mathbf{X}\mathbf{u} & = \mathbf{F} - \mathbf{I} \\ | |
| \end{align}
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| </math>
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| || or
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| || <math>\begin{align}
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| u_i & = x_i - \delta_{iJ} X_J = x_i - X_i\\
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| \frac{\partial u_i}{\partial X_K} & = \frac{\partial x_i}{\partial X_K}-\delta_{iK} \\
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| \end{align}
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| </math>
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| |}
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| where <math>\mathbf{F}</math> is the ''deformation gradient tensor''.
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| Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the ''spatial displacement gradient tensor'' <math>\ \nabla_{\mathbf x}\mathbf U</math>. Thus we have,
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| :{| | |
| |-
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| | <math> \begin{align}
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| \mathbf U(\mathbf x,t) &= \mathbf x - \mathbf X(\mathbf x,t) \\
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| \nabla_{\mathbf x}\mathbf U &= \mathbf I - \nabla_{\mathbf x}\mathbf X \\
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| \nabla_{\mathbf x}\mathbf U &= \mathbf I -\mathbf F^{-1}\\
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| \end{align}
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| </math>
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| || or
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| || <math>\begin{align}
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| U_J& = \delta_{Ji}x_i-X_J =x_J - X_J\\
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| \frac{\partial U_J}{\partial x_k} &= \delta_{Jk}-\frac{\partial X_J}{\partial x_k}\\
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| \end{align}
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| </math>
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| |}
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| ==Examples of deformations==
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| Homogeneous (or affine) deformations are useful in elucidating the behavior of materials. Some homogeneous deformations of interest are
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| * uniform extension
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| * pure dilation
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| * simple shear
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| * pure shear
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| Plane deformations are also of interest, particularly in the experimental context.
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| ===Plane deformation===
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| A plane deformation, also called ''plane strain'', is one where the deformation is restricted to one of the planes in the reference configuration. If the deformation is restricted to the plane described by the basis vectors <math>\mathbf{e}_1, \mathbf{e}_2</math>, the [[deformation gradient]] has the form
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| :<math>
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| \boldsymbol{F} = F_{11}\mathbf{e}_1\otimes\mathbf{e}_1 + F_{12}\mathbf{e}_1\otimes\mathbf{e}_2 + F_{21}\mathbf{e}_2\otimes\mathbf{e}_1 + F_{22}\mathbf{e}_2\otimes\mathbf{e}_2 + \mathbf{e}_3\otimes\mathbf{e}_3
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| </math>
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| In matrix form,
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| :<math>
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| \boldsymbol{F} = \begin{bmatrix} F_{11} & F_{12} & 0 \\ F_{21} & F_{22} & 0 \\ 0 & 0 & 1 \end{bmatrix}
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| </math>
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| From the [[polar decomposition theorem]], the deformation gradient, up to a change of coordinates, can be decomposed into a stretch and a rotation. Since all the deformation is in a plane, we can write<ref name=Ogden/>
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| :<math>
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| \boldsymbol{F} = \boldsymbol{R}\cdot\boldsymbol{U} =
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| \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}
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| \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & 1 \end{bmatrix}
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| </math>
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| where <math>\theta</math> is the angle of rotation and <math>\lambda_1</math>,<math>\lambda_2</math> are the [[finite strain theory|principal stretches]].
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| ====Isochoric plane deformation====
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| If the deformation is isochoric (volume preserving) then <math>\det(\boldsymbol{F}) = 1</math> and we
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| have | |
| :<math>
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| F_{11} F_{22} - F_{12} F_{21} = 1
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| </math>
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| Alternatively,
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| :<math>
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| \lambda_1\lambda_2 = 1
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| </math>
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| ====Simple shear====
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| A simple shear deformation is defined as an isochoric plane deformation in which there are a set of line elements with a given reference orientation that do not change length and orientation during the deformation.<ref name=Ogden/>
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| If <math>\mathbf{e}_1</math> is the fixed reference orientation in which line elements do not deform during the deformation then <math>\lambda_1 = 1</math> and <math>\boldsymbol{F}\cdot\mathbf{e}_1 = \mathbf{e}_1</math>.
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| Therefore,
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| :<math>
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| F_{11}\mathbf{e}_1 + F_{21}\mathbf{e}_2 = \mathbf{e}_1 \quad \implies \quad F_{11} = 1 ~;~~ F_{21} = 0
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| </math>
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| Since the deformation is isochoric,
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| :<math>
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| F_{11} F_{22} - F_{12} F_{21} = 1 \quad \implies \quad F_{22} = 1
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| </math>
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| Define <math>\gamma := F_{12}\,</math>. Then, the deformation gradient in simple shear can be expressed as
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| :<math>
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| \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
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| </math>
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| Now,
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| :<math>
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| \boldsymbol{F}\cdot\mathbf{e}_2 = F_{12}\mathbf{e}_1 + F_{22}\mathbf{e}_2 = \gamma\mathbf{e}_1 + \mathbf{e}_2
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| \quad \implies \quad
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| \boldsymbol{F}\cdot(\mathbf{e}_2\otimes\mathbf{e}_2) = \gamma\mathbf{e}_1\otimes\mathbf{e}_2 + \mathbf{e}_2\otimes\mathbf{e}_2
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| </math>
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| Since <math>\mathbf{e}_i\otimes\mathbf{e}_i = \boldsymbol{\mathit{1}}</math> we can also write the deformation gradient as
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| :<math>
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| \boldsymbol{F} = \boldsymbol{\mathit{1}} + \gamma\mathbf{e}_1\otimes\mathbf{e}_2
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| </math>
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| ==See also==
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| * [[Euler–Bernoulli beam theory]]
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| * [[Deformation (engineering)]]
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| * [[Finite strain theory]]
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| * [[Infinitesimal strain theory]]
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| * [[Moiré pattern]]
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| * [[Shear modulus]]
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| * [[Shear stress]]
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| * [[Shear strength]]
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| * [[Stress (mechanics)]]
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| * [[Stress measures]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{cite book
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| | last = Bazant|first = Zdenek P.
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| | coauthors = Cedolin, Luigi
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| | title = Three-Dimensional Continuum Instabilities and Effects of Finite Strain Tensor, chapter 11 in “Stability of Structures”, 3rd ed.
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| | publisher = World Scientific Publishing
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| | year = 2010
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| | location = Singapore, New Jersey, London
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| | url = http://books.google.com/books?id=qwPlsZF6thUC
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| | isbn = 9814317039}}
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| | |
| *{{cite book
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| | last = Dill|first = Ellis Harold
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| | title =Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity
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| | publisher = CRC Press
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| | year = 2006
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| | location = Germany
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| | url = http://books.google.com/?id=Nn4kztfbR3AC
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| | isbn = 0-8493-9779-0}}
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| | |
| *{{cite book
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| | last = Hutter|first = Kolumban
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| | coauthors = Klaus Jöhnk
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| | title = Continuum Methods of Physical Modeling
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| | publisher = Springer
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| | year = 2004
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| | location = Germany
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| | url = http://books.google.ca/books?id=B-dxx724YD4C
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| | isbn = 3-540-20619-1}}
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| | |
| *{{cite book
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| | last = Jirasek|first = M
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| | coauthors = Bazant, Z.P.
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| | title =Inelastic Analysis of Structures
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| | publisher = J. Wiley & Sons
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| | year = 2002
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| | location = London and New York
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| | url = http://books.google.com/books/?id=8mz-xPdvH00C
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| | isbn = 0471987166}}
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| | |
| *{{cite book
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| | last = Lubarda
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| | first = Vlado A.
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| | title = Elastoplasticity Theory
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| | publisher = CRC Press
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| | year = 2001
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| | url = http://books.google.ca/books?id=1P0LybL4oAgC
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| | isbn = 0-8493-1138-1}}
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| | |
| *{{cite book
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| | last = Macosko
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| | first = C. W.
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| | authorlink =
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| | coauthors =
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| | title = Rheology: principles, measurement and applications
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| | publisher = VCH Publishers
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| | year = 1994
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| | isbn = 1-56081-579-5}}
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| *{{cite book
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| | last = Mase
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| | first = George E.
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| | title = Continuum Mechanics
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| | publisher = McGraw-Hill Professional
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| | year = 1970
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| | url = http://books.google.com/?id=bAdg6yxC0xUC
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| | isbn = 0-07-040663-4}}
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| *{{cite book
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| | last = Mase
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| | first = G. Thomas
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| | coauthors = George E. Mase
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| | title = Continuum Mechanics for Engineers
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| | publisher = CRC Press
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| | year = 1999
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| |edition= Second
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| | url = http://books.google.com/?id=uI1ll0A8B_UC
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| | isbn = 0-8493-1855-6}}
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| | |
| *{{cite book
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| | last = Nemat-Nasser
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| | first = Sia
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| | title = Plasticity: A Treatise on Finite Deformation of Heterogeneous Inelastic Materials
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| | publisher = Cambridge University Press
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| | year = 2006
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| | location = Cambridge
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| | url = http://books.google.ca/books?id=5nO78Rt0BtMC
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| | isbn = 0-521-83979-3}}
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| | |
| *{{cite book
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| | last = Prager|first = William
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| | title =Introduction to Mechanics of Continua
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| | publisher = Ginn and Co.
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| | year = 1961
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| | location = Boston
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| | url = http://books.google.com/books/?id=Feer6-hn9zsC
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| | isbn = 0486438090}}
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| {{DEFAULTSORT:Deformation (Mechanics)}}
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| [[Category:Tensors]]
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| [[Category:Continuum mechanics]]
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| [[Category:Non-Newtonian fluids]]
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| [[Category:Solid mechanics]]
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| [[Category:Deformation]]
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